Diagnose Solr connectivity issues on production

This blog post is for you if you find the following warnings in Sitecore logs:

[Index=<indexname>] Crawling Paused
[Index=<indexname>] Synchronous Indexing Strategy is disabled while indexing is paused.
IndexingStateSwitcher: Indexing is not resumed since indexes have not been initialized yet…
Exception: SolrNet.Exceptions.SolrConnectionException, Message: Unable to connect to the remote server